Parameter Sensitivity Analysis of an Industrial Synchronous Generator Attila Fodor, Attila Magyar, Katalin M. Hangos Abstract A previously developed simple dynamic model of an industrial size synchronous generator operating in a nuclear power plant is analyzed in this paper. The constructed statespace model consists of a nonlinear state equation and a bilinear output equation. It has been shown that the model is locally asymptotically stable with parameters obtained from the literature for a similar generator. The effect of load disturbances on the partially controlled generator has been analyzed by simulation using a PI controller. It has been found that the controlled system is stable and can follow the set-point changes in the effective power well. The sensitivity of the model for its parameters has also been investigated and parameter groups have been defined according to the system s degree of sensitivity to them. This groups form the different candidates of parameters for a subsequent parameter estimation. I. INTRODUCTION Nuclear power plants generate electrical power from nuclear energy, where the final stage of the power production includes a synchronous generator that is driven by a turbine. Similarly to other power plants, both the effective and reactive components of the generated power depend on the need of the consumers and on their own operability criteria. This consumer generated time-varying load is the major disturbance that should be taken care of by the generator controller. The turbo generator, the subject of our study, is a specific synchronous generator with a special cooling system. The armature has been cooled by water and the rotor has been cooled by hydrogen. In the examined nuclear power station the exciter field regulator of the synchronous generator currently does not control the reactive power, only the effective power. The final aim of our study is to design a controller that can control the reactive power such that its generation is minimized in such a way that the quality of the control of the effective power remains (nearly) unchanged. Because of the specialities and great practical importance of the synchronous generators in power plants, their modeling for control purposes is well investigated in the literature. We acknowledge the financial support of this work by the Hungarian State and the European Union under the TAMOP-4.2.1/B-9/1/KONV-21-3 project also this work was supported by the Hungarian Research Fund through grant K67625 A. Fodor is with Department of Electrical Engineering and Information Systems, University of Pannonia, 82 Veszprém, Hungary foa@almos.vein.hu A. Magyar is with Department of Electrical Engineering and Information Systems, University of Pannonia, 82 Veszprém, Hungary amagyar@almos.vein.hu K. M. Hangos is with Department of Electrical Engineering and Information Systems, University of Pannonia, 82 Veszprém, Hungary, and with the Process Control Research Group, Computer and Automation Research Institute HAS, Budapest, Hungary hangos@sztaki.hu Besides of the basic textbooks (see e.g. 1 and 2), there are papers that describe the modeling and use the developed models for the design of various controllers 3, 4. These papers, however, do not take the special circumstances found in nuclear power plants into account that may result in special generator models. The aim of this paper is to perform model verification and parameter sensitivity analysis of a simple dynamic model of a synchronous generator proposed in 5. The result of this analysis will be the basis of a subsequent parameter estimation step. II. THE MODEL OF THE SYNCHRONOUS GENERATOR In this section the bilinear state-space model for a synchronous generator is presented 5 that will be used for local stability analysis and parameter sensitivity analysis. A. Modeling assumptions For constructing the synchronous generator model, let us make the following assumptions: a symmetrical tri-phase stator winding system is assumed, one field winding is considered to be in the machine, there are two amortisseur or damper windings in the machine, all of the windings are magnetically coupled, the flux linkage of the windings is a function of the rotor position, the copper losses and the slots in the machine are neglected, the spatial distribution of the stator fluxes and apertures wave are considered to be sinusoidal, stator and rotor permeability are assumed to be infinite. It is also assumed that all the losses due to wiring, saturation, and slots can be neglected. The six windings (three stators, one rotor and two damper) are magnetically coupled. Since the magnetic coupling between the windings is a function of the rotor position, the flux linking of the windings is also a function of the rotor position. The actual terminal voltage v of the windings can be written in the form J J v ± (r j i j ) ± ( λ j ), j1 j1 where i j are the currents, r j are the winding resistances, and λ j are the flux linkages. The positive directions of the stator currents point out of the synchronous generator terminals. Thereafter, the two stator electromagnetic fields, both traveling at rotor speed, were identified by decomposing each
stator phase current under steady state into two components, one in phase with the electromagnetic field and an other phase shifted by 9 o. With the aboves, one can construct an airgap field with its maxima aligned to the rotor poles (d axis), while the other is aligned to the q axis (between poles) (see Fig. 1). B. Flux linkage equations The generator consists of six coupled coils referred to with indices a, b, c (the stator phases coils), F, D, and Q (the field coil, the d-axis amortisseur and the q-axis amortisseur). The linkage equations are in the following form: T λa λ b λc λ F λ D λ Q Laa L ab Lac L af L ad L aq L ba L bb L bc L bf L bd L bq Lca L cb Lcc L cf L cd L cq L F a L F b L F c L F F L F D L F Q L Da L Db L Dc L DF L DD L DQ L Qa L Qb L Qc L QF L QD L QQ ia i b ic i F i D iq (3) where L xy is the coupling inductance of the coils. It is important to note that the inductances are time varying since Θ is a function of time. The time varying inductances can be simplified by referring all quantities to a rotor frame of reference through Park s transformation: P λabc P I 3 λ FDQ I 3 Laa LaR P 1 P iabc, L Ra L RR I 3 I 3 i FDQ (4) Fig. 1. The abc and dq frames of the generator This method is called the Park s transformation that gives the following relationship: i dq P i abc i abc P 1 i dq (1) where the current vectors are i dq T i i d i q and i abc T i a i b i c and the Park s transformation matrix is: P 2 3 1 2 1 2 1 2 ia cos(θ) i b cos(θ 2π 3 ) ic cos(θ 4π 3 ) ia sin(θ) i b sin(θ 2π 3 ) ic sin(θ 4π 3 ) where i a, i b and i c are the phase currents and Θ rad is the angle between the phase current i a and the current i d. Park s transformation uses three variables: d and q axis components (i d and i q ) and stationary current component (i ), which is proportional to the zero-sequence current. All flux components correspond to an electromagnetic field (EMF), the generator EMF is primarily along the rotor q axis. The angle between this EMF and the output voltage is the machine torque angle δ, where the phase a is the reference voltage of the output voltage. The position of the d axis (in radian) is Θ ω r t δ π/2, where ω r is the rated synchronous angular frequency. Finally, the the voltage and linkage equations are v dq P v abc and λ dq P λ abc, where the vectors are v dq v v d v q T and v abc v a v b v c T, and the linkage flux vectors are λ dq λ λ d λ q T and λabc λ a λ b λ c T. The value of the active power can be written (using (1)) in both coordinate systems: p v T abci abc v T dqpp 1 i dq v T dqi dq (2) where L RR is the rotor-rotor, L aa is the stator-stator, L ar and L Ra are the stator-rotor inductance matrices. P is the Park s transformation matrix, I 3 is the 3 3 unit matrix. The obtained transformed flux linkage equations are as follows: T λ λ d λq λ F λ D λ Q L L d km F km D Lq km Q km F L F M R km D M R L D km Q L Q where: L d L s M s 3 2 L m L L s 2M s k C. Voltage equations i i d iq i F i D iq L q L s M s 3 2 L m 2 3 (5) The schematic equivalent circuit of the synchronous machine can be seen in Fig. 2, and the voltage equations (7) can be derived from it. Fig. 2. vabc The simplified schema of the synchronous machine v FDQ Rabc iabc R FDQ i FDQ λ abc vn, (6) (7)
where R abc diag( r a r b r c ), and RFDQ diag( r F r D r Q ). The neutral voltage v n can also be deduced from Fig. 2 as follows: vn Rni abc Lnm i abc, (8) where L nm L n U 3, and R n r n U 3, and U 3 denotes the 3 3 matrix of full ones. The direct, quadratic, field and amortisseur component of the voltage using Park s transformation: P I 3 vabc v FDQ v dq v FDQ Using (1), it is possible to expand the voltages of the resistances from (9) as P Rabc iabc I 3 R FDQ i FDQ PR abc P 1 ˆR abc i dq. i FDQ i dq i FDQ (9) Finally, the neutral voltage is derived as v dq v FDQ λdq R dq ṖP 1 λ dq ndq i dq i FDQ where n dq is the voltage drop from the neutral network. n dq Pvn PRnP 1 Pi abc PLnmP 1 P i abc 3rni 3Ln i PRnP 1 i dq PLnmP 1 i dq In balanced condition the v voltage is. The above equation can be written in the following form: T v dq v FDQ R i dq R R i FDQ λ dq S ndq where R diag( r r ), R R diag( r F r D r Q ), and S ωλq ωλ d T We can write the voltage equation in simplified matrix form as v dfdqq R RSω i dfdqq L i dfdqq, (14) (11) (12) (13) where v dfdqq v d v F v D v q v Q T, i dfdqq T i d i F i D i q i Q while RRSω and L are the following expressions R RSω L r ωlq ωkm Q r F r D ωl d ωkm F ωkm D r r Q Ld km F km D km F L F M R km D M R L D Lq km Q km Q L Q The state-space model for the currents is obtained by expressing i dfdqq from (14), i.e. i dfdqq L 1 R RSω i dfdqq L 1 v dfdqq (15) D. Mechanical equation The next step is to derive the mechanical part of the model 2. The energy balance is written in the form Fig. 3. The simplified equivalent circuit of the transformed stator and rotor circuits Using the initial assumption of symmetrical tri-phase stator windings (i.e. r a r b r c r), we obtain ˆR abc R abc diag( r r r ). The time derivatives of the fluxes can be computed similarly where and the last term is P λabc I 3 P λ abc λ dq Ṗλ abc λ dq ṖP 1 λ dq, ṖP 1 λ dq ω 1 1 λ λ d λq P λ abc, (1) ωλq dw out dw Mech dw F ield dw Ω, (16) where W Ω is the energy losses in the resistance of the machine, W F ield is the energy of the field, W Mech is the mechanical energy and W out is the output energy of the synchronous generator. The time derivative of (16) is the power equation: dw out dt dw Mech dt dw F ield dt dw Ω dt (17) p out p Mech p F ield p Ω (18) On the other hand, using (2), the output power of tri-phase system is: p out v T abci abc v T dqi dq (19)
The mechanical torque (T Mech ) is obtained by dividing power by the angular velocity ω dθ dt, i.e. T Mech P Mech ω. This gives T Mech λ d i q λ q i d (2) From Newton s second law the equation of motion is 2H ω B ω T Mech T Electr T Dump, (21) where T Mech is the mechanical torque, T Electr is the electrical torque per phase, H is the inertia constant and T Dump is the dumping torque. The time and the rotation speed using per units i.e. dimensionless variables is t u ω B t and ω u ω/ω B. Afterwards the normalized swing equation can be written as 2Hω B dω u dt u T Mech T Electr T Dump, (22) where 2Hω B τ j, where τ j is a time-like quantity coming from per unit notation not detailed here. The electrical torque can be expressed from the flux and the current of the machine T Electr 1 3 (λ di q λ q i d ) (23) It is often convenient to write the damping torque as T Dump Dω, where D is a damping constant. The electrical torque expressed using the state vector of model (15) is then 3 T Elect Ld iq km F iq km D iq Lq i d km Q i d T id if i D iq i Q Since the variables have a few orders of magnitude difference in their values in natural units, the equations are normalized with respect to a base value (corresponding to the normal range of the variables). This way all signals are measured in normalized units (p.u.). Using (22) and the definition of τ j, the speed of the synchronous machine is ω L d iq km F iq km D iq Lq i d km Q i d D τ j T i d i F i D iq i Q ω (24) T Mech τ j (25) Note, that (25) can be used as an additional state equation for state space model (15). The loading angle (δ) of the synchronous generator is t δ δ (ω ω r )dt t that we can differentiate to obtain the time derivative of the δ in per unit notation δ ω 1, (26) so the loading angle (δ) can also be regarded as a state variable in the state space model (15, 25). Altogether, there are 6 state variables: i d, i F, i D, i q, i Q, ω and δ. The input variables (i.e. manipulatable inputs and disturbances) are: T Mech, v F, v d and v q. Observe, that the state equations (15, 25, 26) are bilinear in the state variables because matrix R RSω in (15) depends linearly on ω. E. Output equations The output active power equation can be written in the following form: p out v d i d v q i q v i (27) Assuming steady-state for the stationary components (v i ), (27) simplifies to p out v d i d v q i q, (28) and the reactive power is q out v d i q v q i d. (29) Equations (28-29) are the output equations of the generator s state-space model. Observe, that these equations are bi-linear in the state and input variables. F. Connecting the synchronous generator to an infinite huge network Since every synchronous machine is connected to an infinite bus (shown in Fig. 4.) the next task is to extend the previous models with an infinite bus. In Fig. 4, resistance R e and inductance L e represent the output transformer of the synchronous generator and the transmission-line. Fig. 4. Synchronous machine connected to an infinite bus The matrix form of the modified voltage equation is as follows: v abc v abc Re I 3 i abc Le I 3 iabc (3) Equation (3) can be transformed to the dq coordinate system as v dq Pv abc Pv abc Re I 3 i dq Le I 3 i dq (31) The tri-phase voltage of the bus in the dq coordinate system is then v dq Pv abc 3V sin(δ α) cos(δ α) Afterwards, one can express the current vector i dq and voltage vector v dq as and P i abc i dq Ṗi abc i dq ṖP 1 i dq (33) v dq v 3 sin(δ α) cos(δ α) Rei dq Le i dq ωle iq The integration of resistance R e and inductance L e into voltage equation (14) can be done by a simple change in matrices R RSω and L. The obtained voltage equation is: v dfdqq R RSω i dfdqq L i dfdqq, (35) where v dfdqq, i dfdqq, i dfdqq R RSω and L are T v dfdqq v d v F v D vq v Q T i dfdqq i d i F i D iq i Q R RSω R RSω diag( Re Re ) L L diag( Le Le ) (32) (34)
From (35) it is possible to express i dfdqq as i dfdqq L 1 R RSω i dfdqq L 1 v dfdqq (36) Now the extended state space model consists of Equations (36, 25, 26). III. MODEL ANALYSIS The above model has been verified by simulation against engineering intuition using parameter values of a similar generator taken from the literature. After the basic dynamical analysis, the set of model parameters is partitioned based on the model s sensitivity on them. A. Generator parameters The parameters are described only for phase a since the machine is assumed to have symmetrical tri-phase stator windings system. The stator mutual inductances for phase a are L ab L ba M s L m cos(2(θ π )) 6 (37) where M s is a given constant. The rotor mutual inductances are L F D L DF M R, L F Q L QF L DQ L QD The phase a stator to rotor mutual inductances are given by: (from phase windings to the field windings) L af L F a M F cos(θ) (38) where the parameter M F is given. The stator to rotor mutual inductance for phase a (from phase windings to the direct axis of the damper windings) is L ad L Da M D cos(θ) (39) with a given parameter M D. The phase a stator to rotor mutual inductances are given by: (From phase windings to the damper quadratic direct axis) L aq L Qa M Q cos(θ) (4) Parameters L d, L q, M D, M F, M R and M D used by the state space model (15, 25, 26, 28,29) are defined as L d L s M s 3 2 L m L q L s M s 3 2 L m M D L AD k M F LAD k M R L AD M Q L AQ k 2 3 k (41) Figure 3 shows the position of the following (inductance and resistance) parameters in the simplified electrical circuit model: L F, L D, L Q, L d, L q, M F, M D, M Q, r F, r D, r Q, Using the initial assumption of symmetrical tri-phase stator windings (i.e. r a r b r c r) we get the resistance of stator windings of the generator. The r F represent the resistance of the rotor windings. The r D and r Q represent the resistance of the d and q axis circuit. The resistance R e and inductance L e represent the output transformer of the synchronous generator and the transmission-line. The parameters of the synchronous generator were obtained from the literature 1. The stator base quantities, the rated power, output voltage, output current and the angular frequency are: S B 16 MVA/3 53.333 MVA V B 15 kv/ 3 8.66 kv I B 6158 A ω e 2πf rad/s The parameters of the synchronous machine and the external network in per units are: L d 1.7 l d.15 L MD.2838 L q 1.64 l q.15 L MQ.2836 L D 1.65 l F.11 r.196 L Q 1.526 l D.55 r F.74 L AD 1.55 l Q.36 r D.131 L AQ 1.49 r Q.54 R e.2 V.828 L e 1.64 D 2.4 B. Local stability analysis The steady-state values of the state variables can be obtained from the steady-state version of state equations (36, 25, 26) using the above parameters. Equation (14) implies that the expected value to i D and i Q are, that coincide with the engineering intuition. The equilibrium point of the system is: ω.999691 i d 1.913269 i q.66751 i F 2.97899982 i D 8.6242856 1 9 i Q 5.3334899 1 1 The state matrix A of the locally linearized state-space model ẋ Ax Bu has the following numerical value in this equilibrium:.361.4.142 3.4851 2.5455 2.3285.124.49.772 1.211.8773.825.228.44.964 2.257 1.611 1.4737 3.5855 2.6464 2.6464.361.91 1.247 3.59 2.5839 2.5839.352.1234 1.5 8 1 6.2.2.8.5.11 The eigenvalues of the state matrix are: λ 1,2 3.61988 1 2 ± j.99774 λ 3.124 λ 4 1.67235 1 3 λ 5 4.724291e 1 4 λ 6.123426 It is apparent that the real part of the eigenvalues are negative but their magnitudes are small, thus the system is on the boundary of the stability domain.
Fig. 5. The controller for the nonlinear model implemented in Matlab/Simulink C. PI controller The control scheme of the synchronous machine is a classical PI controller (Fig 5) that ensures stability of the equilibrium point under small perturbations 4. The controlled output is the speed (ω), the manipulated input is the mechanical torque T Mech. The proportional parameter of the PI controller of the speed is.5 and the integrator time is.1 in per units. D. Model validation The dynamic properties of the generator have been investigated in such a way that a single synchronous machine was connected to an infinite bus that models the electrical network (see Fig. 4). The response of the speed controlled generator has been tested under step-like changes of the exciter voltage. The simulation results are shown in Fig. 6, where the exciter voltage v F and the torque angle δ are shown. When the exciter voltage is increased the loading angle must be decreased as it can be seen in the Fig. 6. E. Sensitivity analysis The aim of this section define parameter groups according to the system s sensitivity on them. Linkage inductances l d, l q, l D, l Q, L MD and L MQ are not used by the current model, only by the flux model 5. It is not expected that the output and the state variables of system change when these parameters are perturbed, see Fig. 7. As it was expected, the model is insensitive for these parameters. Note, that the linkage inductance parameters are only used for calculating the fluxes of the generator. Fig. 6. Response to the exciter voltage step change of the controlled generator ( means the deviation form the steady-state value) Sensitivity of the model to the controller parameters P and I and the dumping constant D has also been investigated. Since the PI controller controls for ω modifying the value of D the controller keeps ω at synchronous speed. This is why the output and the steady state value of the system variables do not change (as it is apparent in Fig 8) even
Fig. 7. The model states and outputs for a ±2% change of l d Fig. 8. The model states and outputs for a ±9% change of D for a considerably large change of D. Sensitivity analysis of the resistance of the stator and the resistance of the transmission line led to the same result. A ±2% perturbation in them resulted a small change in currents i d, i q and i F, this causes the change of the effective and the reactive power of the generator, as it is shown in Fig. 9. The analysis of the rotor resistance r F showed, that the ±2% perturbation of r F kept the quadratic component of the stator current (i q ) constant, but currents i d and i F were changed. The output of the generator also changed, as it is shown in Fig. 1. The sensitivity of the model states and outputs to the inductance of the rotor (L F ) and the inductance of the direct axis (L D ) has also been analyzed. The results shows only a moderate reaction in i d and i F to the parameter perturbations, and the equilibrium state of the system kept unchanged. However, decreasing the value of the parameters to the 9 percent of their nominal value destabilized the system. The results of a ±9% perturbation in L F are shown in Fig. 11. A small perturbation of the outputs is noticeable. Finally, the sensitivity of the model (15, 25, 26, 28,29) to the linkage inductance L AD has been examined. When the parameter has been changed ±5%, currents i d and i F changed only a little. On the other hand, the steady-state of the system has shifted as it can be seen in Fig. 12. A parameter variation of 5% destabilized the system. As a result of the sensitivity analysis, it is possible to define the following groups of parameters: Not sensitive inductances l d, l q, l D, l Q, L MD, L MQ, L AQ, L Q, damping constant D and the controller parameters P and I. Since the state space model of interest is insensitive for them, the values of these parameters cannot be determined from measurement data using any parameter estimation method. Sensitive Less: resistances of the stator r and the transmission-line R e. More: resistance r F of the rotor and the inductance of transmission-line L e. These parameters are candidates for parameter estimation. Critically sensitive linkage inductance L AD, inductances L D and L F. These parameters can be estimated very well.
Fig. 9. The model states and outputs for a ±2% change of r resist Fig. 1. The model states and outputs for a ±2% change of r F IV. CONCLUSION AND FURTHER WORK The simple bilinear dynamic model of an industrial size synchronous generator operating in a nuclear power plant described in 5 has been investigated in this paper. It has been shown that the model is locally asymptotically stable around a physically meaningful equilibrium state with parameters obtained from the literature for a similar generator. The effect of load disturbances on the partially controlled generator has been analyzed by simulation by using a traditional PI controller. It has been found that the controlled system is stable and can follow the setpoint changes in the effective power well. Eighteen parameters of the system has been selected for sensitivity analysis, and the sensitivity of the state variables and outputs has been investigated for all of them. As a result, the parameters has been partitioned to four groups. Based on the results presented here, the further aim of the authors is to estimate the parameters of the model for a real system from measurements. The sensitivity analysis enables us to select the candidates for estimation that are r F, L e, L AD, L D and L F. supported by the Hungarian Research Fund through grant K67625. REFERENCES 1 P.M. Anderson and A.A. Fouad: Power-Systems-Control and Stability, The IOWA State University Press, Ames Iowa, 1977. 2 P.M. Anderson, B.L. Agrawal, J.E. Van Ness: Subsynchronous Resonance in Power Systems, IEEE Press, New York, 199. 3 A. Loukianov, J.M. Canedo, V.I. Utkin and J. Cabrera-Vazquez: Discontinuous controller for power systems: sliding-mode bock control approach. IEEE Trans. on Industrial Electronics, 51:34-353, 24 4 F.P. de Mello and C. Concordia: Concepts of synchronous machine stability as affected by excitation control, IEEE Trans. Power Application Systems, PAS-88:316-329, 1969. 5 Attila Fodor, Attila Magyar, Katalin M. Hangos: Dynamic Modeling and Model Analysis of a Large Industrial Synchronous Generator Proc. of Applied Electronics 21, Pilsen, Czech Republic (accepted) V. ACKNOWLEDGEMENT We acknowledge the financial support of this work by the Hungarian State and the European Union under the TAMOP- 4.2.1/B-9/1/KONV-21-3 project also this work was
Fig. 11. The model states and outputs for a ±9% change of LF Fig. 12. The model states and outputs for a ±4% change of LAD